A Popular Website Requires Users to Create a Password Consisting of Digits Only GMAT Problem Solving

Question: A popular website requires users to create a password consisting of digits only. If no digit may be repeated and each password must be at least 9 digits long, how many passwords are possible.

1. 9! + 10!
2. 2*10!
3. 9! * 10!
4. 19!
5. 20!

Answer and Solution:

Approach Solution:

There is only one approach to this question.

Explanation: Given to us that a popular website requires users to create a password consisting of digits only. No digit can be repeated in the passwords and each password must be at least 9 digits long. It has asked to find out the number of possible passwords.
It should be noted that there are 10 possible digits –
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
It is given that a password is at least 9 digits long, so it can contain 9 of these 10 digits leaving one digit unused or all the 10 digits.

This is a question from permutation and combination. All the formulas required to solve this problem are mentioned below-
Permutation of n objects taking r at a time = \(=^nP_r=n!/(n-r)!\)
Combination of n objects taking r at a time =\(=^nC_r=n!/((n-r)!*r!)\)
Also \(^nP_r=^nC_r*r!\)
Where n! = 1*2*3*4……..(n-1) * n

Here in this question, we have to find all the permutations of 10 digits to form a nine-digit number.
It should be noted that when forming a 9-digit number we cannot place zero in the first place but in the case of passwords it is also feasible to put 0 in the first place.
The total number of permutations to make 9-digit passwords from 10 digits will be

\(^{10}P_9=10!/(10-9)!=10!/1!=10!\)

There is one more possibility where the password can contain all the 10 digits.
The total number of permutations to make 10-digit passwords from 10 digits will be

\(^{10}P_{10}=10!/(10-10)!=10!/0!\)
= 10!

Total number of ways to create passwords containing at least nine digits = total number of permutations to make 9-digit passwords + the total number of permutations to make 10-digit passwords
Total permutation = 10! + 10!
= 2 * 10!

Therefore the correct option is option B.

Correct Answer: B

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